$Z_c(4430)$, $Z_c(4200)$, $Z_1(4050)$, and $Z_2(4250)$ as triangle singularities

Abstract

$Z_c(4430)$ discovered in $\bar{B}^0\to\psi(2S)K^-\pi^+$, $Z_c(4200)$ found in $\bar{B}^0\to J/\psi K^-\pi^+$, and $Z_1(4050)$ and $Z_2(4250)$ observed in $\bar{B}^0\to\chi_{c1}K^-\pi^+$ are candidates of charged charmonium-like states. All surviving theoretical models interpreted these candidates as four-quark states, until we recently identified a compelling alternative. We discuss that kinematical singularities in triangle loop diagrams induce a resonance-like behavior that can consistently explain the properties (such as spin-parity, mass, width, and Argand plot) of $Z_c(4430)$, $Z_c(4200)$, $Z_1(4050)$ and $Z_2(4250)$ from experiments. In terms of the triangle singularities, we can also naturally understand interesting experimental findings such as the appearance (absence) of $Z_c(4200)$($Z_c(4430)$)-like contribution in $\Lambda_b^0\to J/\psi p\pi^-$, and the highly asymmetric shape of the spectrum bump for $Z_1(4050)$; the other theoretical models have not successfully addressed these points. Although Pakhlov et al. proposed another triangle diagram to generate a $Z_c(4430)$-like bump, we argue that this scenario is very unlikely.